If for some $\alpha,\beta;\ \alpha\le\beta,\ \alpha+\beta=8$ and $\sec^2( an^{-1}\alpha)+\operatorname{cosec}^2(\cot^{-1}\beta)=36$, then $\alpha^2+\beta$ is ______.

Question

If for some $\alpha,\beta;\ \alpha\le\beta,\ \alpha+\beta=8$ and $\sec^2(	an^{-1}\alpha)+\operatorname{cosec}^2(\cot^{-1}\beta)=36$, then $\alpha^2+\beta$ is ______.

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] $\sec^2(	an^{-1}\alpha)=1+\alpha^2$ and $\operatorname{cosec}^2(\cot^{-1}\beta)=1+\beta^2$, so $\alpha^2+\beta^2=34$. With $\alpha+\beta=8$: $\alpha\beta=\frac{64-34}2=15$, so $\alpha=3,\ \beta=5$. $\alpha^2+\beta=9+5=14$. (Printed signs in the question read as $=$ and $+$.) Answer: 14.

Q. If for some $\alpha,\beta;\ \alpha\le\beta,\ \alpha+\beta=8$ and $\sec^2(\tan^{-1}\alpha)+\operatorname{cosec}^2(\cot^{-1}\beta)=36$, then $\alpha^2+\beta$ is ______.

Explanation

Reviewed by Dr. Abubacker

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